Top PDF On transitioning from type-1 to interval type-2 fuzzy logic systems

On transitioning from type-1 to interval type-2 fuzzy logic systems

On transitioning from type-1 to interval type-2 fuzzy logic systems

Abstract-Capturing the uncertainty arising from system noise has been a core feature of fuzzy logic systems (FLSs) for many years. This paper builds on previous work and explores the methodological transition of type-l (Tl) to interval type-2 fuzzy sets (IT2 FSs) for given "levels" of uncertainty. Specifically, we propose to transition from Tl to IT2 FLSs through varying the size of the Footprint Of Uncertainty (FOU) of their respective FSs while maintaining the original FS shape (e.g., triangular) and keeping the size of the FOU over the FS as constant as possible. The latter is important as it enables the systematic relating of FOU size to levels of uncertainty and vice versa, while the former enables an intuitive comparison between the Tl and T2 FSs. The effectiveness of the proposed method is demonstrated through a series of experiments using the well-known Mackey-Glass (MG) time series prediction problem. The results are compared with the results of the IT2 FS creation method introduced in [1] which follows a similar methodology as the proposed approach but does not maintain the membership function (MF) shape.
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General Derivation And A Design Methodology For Interval Type-2 Fuzzy Logic Systems

General Derivation And A Design Methodology For Interval Type-2 Fuzzy Logic Systems

In comparison with T1FLS, there are few studies which analyze the mathematical closed-form structure of IT2FLS in the literature. The mathematical structure of Mamdani interval type-2 fuzzy PI which uses two triangular type-2 fuzzy sets for each input and four singletons for output has been studied in [24]. The Zadeh AND operator and two different type reducers; namely, the popular centroid and the average defuzzifiers were used in their analysis. The analytical structure of a special class of interval type-2 fuzzy PI and PD controllers that have symmetrical rule-base and symmetrical consequent sets is presented in [25]. The Karnik-Mendel (KM) type reduction [31] and Zadeh AND operator were used in analytical structure of this study. In [25] , it has been shown that the IT2-FLCs partition the input domain into 31 extra local regions in comparison with its type-1 counterpart and each region provides a unique relationship between the inputs and output signals. In the mentioned study, the IT2-FLC has been compared with the corresponding T1-FLC and the potential advantages of using IT2-FLC over type-1 are examined. The classical type-2 MFs and Zadeh AND operator in both of the above studies cause more complexity in mathematical relationship between the inputs and the output of IT2FLS. Therefore, it becomes very difficult to generalize these analyses to IT2FLS with more than two MFs for each input. In practice, most of IT2FLS studies use product AND operator because of a fine performance and simple algorithm that is easy to be implemented. A systematical methodology to construct an IT2-FLC by using diamond-shaped type-2 MFs was proposed in [37]. There, the closed-form relation between input and output of the proposed IT2-FLC was derived which provides a way of understanding why IT2-FLC is more robust and how it copes with uncertainties.
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Learning of interval and general type 2 fuzzy logic systems using simulated annealing: theory and practice

Learning of interval and general type 2 fuzzy logic systems using simulated annealing: theory and practice

fuzzy logic is a growing research topic with much evidence of successful applications. However, almost all developments of type-2 fuzzy logic systems have been based on interval type-2 fuzzy logic [45][27]. The heavy computational load associated with the generalized form of type-2 sets is the main driver for the lack of applications of general type-2 fuzzy sets compared with the interval model. This prior work has reinforced the common concept that interval type-2 fuzzy logic systems can add more modeling capabilities than type-1 fuzzy logic systems but with extra computational cost. Learn- ing and optimization of general type-2 fuzzy logic systems are open areas for more research, as well as the ongoing research on how to reduce the complexity of general type-2 fuzzy logic systems, especially in the type-reduction phase of the system. The large number of methods used to design type-1 and interval type-2 fuzzy logic systems can be seen as potential candidates for general type-2 fuzzy logic systems and some of them might uncover further possibilities for modeling uncertainty. However, recent ad- vances in general type-2 fuzzy logic systems research, including new representations, optimized operations and faster type-reduction methods, indicate an expected growth in applications. Despite the larger number of computations associated with general type- 2 fuzzy sets, there may well be benefits compared to interval type-2 fuzzy sets. This ability can be unveiled using automated designing methods rather than being chosen by the designer manually. Automated methods can fine-tune initial fuzzy logic system designs due to the lack of a rational basis for choosing secondary membership func- tions for general type-2 fuzzy sets [36, p.302]. This issue enforces the need for using automated methods in such problem. The other factor affecting the usage of general type-2 fuzzy logic systems is the lack of practical parameterization methods to handle the third dimension in general type-2 fuzzy sets. In general, a general type-2 fuzzy logic system has the potential to model more uncertainties despite the large amount of computations associated with it especially when applied to non real-time applications. In consequence, the question of how much general type-2 fuzzy logic systems can add to modeling performance over interval type-2 fuzzy logic systems is another issue that warrants investigation.
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Optimizing of Interval Type-2 Fuzzy Logic Systems Using Hybrid Heuristic Algorithm Evaluated by Classification

Optimizing of Interval Type-2 Fuzzy Logic Systems Using Hybrid Heuristic Algorithm Evaluated by Classification

79 We can describe the interval type-2 fuzzy logic system as follows: the crisp sets inputs are first fuzzified into input interval type-2 fuzzy sets. In the fuzzifier, it creates the membership function which consists of types of membership function, linguistic variable and fuzzy rule base. It has many types of the membership function such as triangular membership function, trapezoidal membership function, Gaussian membership function, Smooth Membership Function, Z- membership function and so on. So, the fuzzifier sends the interval type-2 fuzzy set into the inference engine and the rule base to produce output type-2 fuzzy sets. The interval type-2 fuzzy logic system rules will remain the same as in the type-1 fuzzy logic system, but the antecedents and/or consequents will be represented by interval type-2 fuzzy sets. A finite number of fuzzy rules, can be represented as if-then forms, then integrates into the fuzzy rule base. A standard fuzzy rule base is shown below.
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Learning of interval and general type-2 fuzzy logic systems using simulated annealing: theory and practice

Learning of interval and general type-2 fuzzy logic systems using simulated annealing: theory and practice

of type-1 fuzzy logic inappropriate in many cases especially with problems related to inefficiency of performance in fuzzy logic control [21]. Problems related to model- ing uncertainty using membership functions of type-1 fuzzy sets have been recognized early and [50] introduced higher types of fuzzy sets called type-n fuzzy sets including type-2 fuzzy sets [37]. Type-2 fuzzy logic systems have many advantages compared with type-1 fuzzy logic systems, including the ability to handle different types of uncer- tainties and the ability to model problems with fewer rules [21]. Two factors should be considered regarding the the widespread perception that a general type-2 fuzzy logic system should outperform the interval form which also should outperform a type-1 fuzzy logic system [46]. These two factors are the dependence of performance on the choice of the model parameters as well as on the variability of uncertainty within the application [46]. Therefore, a good choice of the model’s parameters using automated methods is desirable to get clearer conclusions regarding this comparison. Despite these promising indicators of the general type-2 fuzzy logic systems, almost all devel- opments of type-2 fuzzy logic systems have been based on interval type-2 fuzzy logic systems. However, new representations allow us to consider general type-2 fuzzy logic systems. These representations include geometric T2FLS [15], alpha-planes [41], al- pha cuts [22] and Z-slices [45, 10][47]. There have been a number of developments in reducing the computations for general type-2 fuzzy logic systems. For type-reduction, the geometric defuzzifier [15], the sampling defuzzifier [19] followed by importance sampling defuzzifier [31] and a centroid defuzzifier based on the alpha representation [32] have been proposed. One attempt to design general type-2 sets based on zSlices representation was proposed in [10] where survey data and device characteristics were used to build zSlices
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Learning of interval and general type-2 fuzzy logic systems using simulated annealing: Theory and practice

Learning of interval and general type-2 fuzzy logic systems using simulated annealing: Theory and practice

Helping develop real-world applications. Another motivation for this research comes from the lack of applications using general type-2 fuzzy logic systems. Type-2 fuzzy logic is a growing research topic with much evidence of successful ap- plications. However, almost all developments of type-2 fuzzy logic systems have been based on interval type-2 fuzzy logic [27,45] . The heavy computational load associated with the generalized form of type-2 sets is the main driver for the lack of applications of general type-2 fuzzy sets compared with the interval model. This prior work has reinforced the common concept that interval type-2 fuzzy logic systems can add more modeling capabilities than type-1 fuzzy logic systems but with extra computational cost. Learning and optimization of general type-2 fuzzy logic systems are open areas for more re- search, as well as the ongoing research on how to reduce the complexity of general type-2 fuzzy logic systems, especially in the type-reduction phase of the system. The large number of methods used to design type-1 and interval type-2 fuzzy logic systems can be seen as potential candidates for general type-2 fuzzy logic systems and some of them might uncover further possibilities for modeling uncertainty. However, recent advances in general type-2 fuzzy logic systems research, including new representations, optimized operations and faster type-reduction methods, indicate an expected growth in applications. Despite the larger number of computations associated with general type-2 fuzzy sets, there may well be benefits compared to interval type-2 fuzzy sets. This ability can be unveiled using automated designing methods rather than being chosen by the designer manually. Automated methods can fine-tune initial fuzzy logic system designs due to the lack of a rational basis for choosing secondary membership functions for general type-2 fuzzy sets [36, p. 302] . This issue enforces the need for using automated methods in such problem. The other factor affecting the usage of general type-2 fuzzy logic systems is the lack of practical parameterization methods to handle the third dimension in general type-2 fuzzy sets. In general, a general type-2 fuzzy logic system has the potential to model more uncertainties despite the large amount of computations associated with it especially when applied to nonreal-time applications. In consequence, the question of how much general type-2 fuzzy logic systems can add to modeling performance over interval type-2 fuzzy logic systems is another issue that warrants investigation.
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Contrasting singleton type 1 and interval type 2 non singleton type 1 fuzzy logic systems

Contrasting singleton type 1 and interval type 2 non singleton type 1 fuzzy logic systems

Moving to the right in Table IV, the FOU size parameter c is increasing and the inputs modelling are transitioning from T1 (c = 0) to IT2 FSs that designed using the proposed FOU creation method with constant FOU size over the core of the primary membership domain. The performance clearly increases with the increased FOU size at each noise level. The reduction of the performance is started at c = 1.0 where the FOU covers the entire primary membership (i.e., LMF is entirely on the primary variable x–axis) [33], and only the UMF of the IT2 input FS is used in the fuzzification and accordingly is considered as T1 non-singleton fuzzification case (see the last column of Table III).
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Application of Type-1 and Type-2 Fuzzy Logic Controller for The Real  Swarm Robot

Application of Type-1 and Type-2 Fuzzy Logic Controller for The Real Swarm Robot

The use of the fuzzy logic method in the navigation task has been analyzed by a lot of previous studies. In overcoming the obstacle avoidance and stabilization of the position of mobile robot wheel problem Faisal et al [32] has designed sensor-based fuzzy sensor wireless for mobile robot navigation tasks between static and moving barriers. While in the paper [33], it can be seen that there were design and implemen- tation of the fuzzy hybrid architecture for intelligent navigation systems and mobile robot control in avoiding obstacles in static and dynamic environments. Just as in the case of robot football, fuzzy logic is very important, applied to individual robot be- haviors and actions, especially for obstacle avoidance and achieving targets [34][35]. In research [8], Algabri et al. have designed two fuzzy logic behaviors for mobile robot navigation i.e., behavior to achieve targets and avoid obstacles with different scenarios. However, it is important to pay attention to the development of this archi- tecture, that is for the same path-planning problem.
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Constrained interval type-2 fuzzy sets

Constrained interval type-2 fuzzy sets

they can maintain the typical interpretability of fuzzy logic systems as long as they contain a reasonably small number of rules and it is possible to give a linguistic label to the MFs involved [34]. The genetic approach proposed for the generation of CIT2 fuzzy systems, is based on the architecture described in [35]. Each of the input variables of the system is partitioned in 3 triangular MFs. The center of each triangular generator set for the antecedent CIT2 FSs is determined using the well known fuzzy C-Means clustering algorithm (FCM) [36] on each input variable. The end-points of the triangles are the center of the previous and next clusters, if they exist, or the closest end-point of the UOD increased by 10% of the UOD size, so that every point in the UOD belongs to at least one of the MFs with a membership value greater than 0. The continuous DS is an interval [−c, c], c > 0 with 2c = 5% of the distance between the starting and end point of each triangular generator set. The output variable is partitioned with a number of CIT2 FS equal of the number of classes in the problem. Each of them is given an integer index from 0 to the number of classes involved. The index represents the peak of their triangular generator set while the start and end point of the triangles are obtained respectively subtracting and adding 1 to their peak points. The DS for all the CIT2 MF partitioning the output is an interval [−c, c], c > 0 with 2c = 10% of the UOD. Once the MFs are determined, there is a first evolutionary stage to generate the rule-base of the system. During this process, the MFs are not changed. The number of rules is fixed (as shown in [35], redundant rules can be eliminated with an additional stage) and each chromosome codes an entire rule-base. With n input variables, each rule is coded with a set of n + 1 integers. Each gene p i represents the index of the MF to use for the i − th antecedent or for the consequent, if i = n + 1. A value of -1 for p i , i ≤ n, indicates that the i − th input must not be included in the rule p i belongs to. A sequence of encoded rules represents a rule-base.
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Interval type–2 fuzzy decision making

Interval type–2 fuzzy decision making

This paper concerns itself with decision making under uncertainty and the consideration of risk. Type-1 fuzzy logic by its (essentially) crisp nature is limited in modelling decision making as there is no uncertainty in the membership function. We are interested in the role that interval type2 fuzzy sets might play in enhancing decision making. Previous work by Bellman and Zadeh considered decision making to be based on goals and constraint. They deployed type1 fuzzy sets. This paper extends this notion to interval type2 fuzzy sets and presents a new approach to using interval type-2 fuzzy sets in a decision making situation taking into account the risk associated with the decision making. The explicit consideration of risk levels increases the solution space of the decision process and thus enables better decisions. We explain the new approach and provide two examples to show how this new approach works.
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Type-2 fuzzy logic system applications for power systems

Type-2 fuzzy logic system applications for power systems

essentially universal approximators suited to model non-linear continuous processes. All these imply that in terms of power systems, suitable scenarios or devices are the ones where processes or tasks are continuous rather that discrete. This is mainly by the fact that FLSs are inherently continuous and discrete problems might imply a information loss in the response discretisation. Besides, type-2 FLSs rather than type-1 are particularly appropriate for problems hard to model because different non-stationary statistical attributes such us inaccuracies in the voltage sensitivity or the noise measurement cannot be expressed ahead of time mathematically for all conditions. One could easily point out, that given the non-discrete nature of many controllers and their non-linear relation with the network’s sensitivity factors, any kind of pre-calculation will lead to inaccuracies. Thus, in those situations where a fast response is needed, computation time can be saved by means of type-2 FLSs. Therefore, their application will probably stand out more clearly in meshed networks rather than radial ones as interactions are less obvious.
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A Comparison of Type-1 and Type-2 Fuzzy Logic Controllers in Robotics: A review

A Comparison of Type-1 and Type-2 Fuzzy Logic Controllers in Robotics: A review

III. Type 2 Fuzzy controllers Type-2 fuzzy logic is a growing research topic—if number of publications is taken as a measure. Key researchers in the fuzzy logic community are now embracing type-2 fuzzy logic and there is much evidence of successful applications, so we can only expect this growth to continue. Other evidence of interest in type-2 fuzzy logic is that there have been special sessions at every Fuzz-IEEE since 1999 where the sessions generally consist of 20 papers or more. Figure 2: The list of reviewed articles related to Type 2 fuzzy controllers in a robotics. Type-2 fuzzy methods provide second order uncertainties allowing fuzzy systems to truly deal with real world uncertainty. In the current climate of ever faster, more powerful and more affordable hardware type-2 fuzzy methods present an exciting opportunity to explore uncertainties in real world.
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A New Controlling Approach of Type 1 Diabetics Based on Interval Type-2 Fuzzy Controller

A New Controlling Approach of Type 1 Diabetics Based on Interval Type-2 Fuzzy Controller

Table 2: AMM Parameters for healthy and diabetics subjects 3 Proposed interval Type -2 Fuzzy Controller Type-2 Fuzzy Logic is an emerging and promising area for achieving Intelligent Control. Using interval type-2 fuzzy logic for minimizing the effects of uncertainty produced by the instrumentation elements environmental noise, etc. A type-2 fuzzy logic system consists basically of three blocks fuzzification, inference and defuzzification as similar to type-1. But the only difference is in the third block of the type-2 fuzzy which is not only defuzzifier but also accomplished by a type-reducer processing block. This difference is mainly associated with the nature of the membership functions where type- reducer is needed due to the added degree in the kind of fuzzy sets. In this article, we proposed the fuzzy controller is structure by singleton fuzziffication and produce the inter-face engine Mamdani and the center of sets method type reducer [33] and KM algorithme for defuzzification. The input variable are the plasma glucose construction and the change rate of error respectively, and the insulin injection rate take into account as the output. Figure1 presents a type 2 fuzzy logic system.
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Variable Impedance Control for Rehabilitation Robot using Interval Type-2 Fuzzy Logic

Variable Impedance Control for Rehabilitation Robot using Interval Type-2 Fuzzy Logic

In this study, a novel variable impedance control for a lower-limb rehabilitation robotic system using voltage control strategy is presented. The majority of existing control approaches are based on control torque strategy, which require the knowledge of robot dynamics as well as dynamic of patients. This requires the controller to overcome complex problems such as uncertainties and nonlinearities involved in the dynamic of the system, robot and patients. On the other hand, how impedance parameters must be selected is a serious question in control system design for rehabilitation robots. To resolve these problems this paper, presents a variable impedance control based on the voltage control strategy. In contrast to the usual current-based (torque mode) the use of motor dynamics lees to a computationally faster and more realistic voltage-base controller. The most important advantage of the proposed control strategy is that the nonlinear dynamic of rehabilitation robot is handled as an external load, hence the control law is free from robot dynamic and the impedance controller is computationally simpler, faster and more robust with negligible tracking error. Moreover, variable impedance parameters based on Interval Type-2 Fuzzy Logic (IT2Fl) is proposed to evaluate impedance parameters. The proposed control is verified by a stability analysis. To illustrate the effectiveness of the control approach, a 1-DOF lower-limb rehabilitation robot is designed. Voltage- based impedance control are simulated through a therapeutic exercise consist of Isometric and Isotonic exercises. Simulation results show that the proposed voltage- based variable impedance control is superior to voltage-based impedance control in therapeutic exercises.
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Learning of Type-2 Fuzzy Logic Systems using Simulated Annealing.

Learning of Type-2 Fuzzy Logic Systems using Simulated Annealing.

Although, type-2 fuzzy logic is a growing research topic with much evidence of successful applications (John and Coupland, 2007), up to now, almost all developments type- 2 fuzzy logic systems were based on interval type-2 fuzzy logic systems with some exceptions related to some works using different representations of type-2 sets and systems such as geometric type-2 fuzzy logic systems (Coupland and John, 2007), alpha- planes (Mendel et al., 2009), alpha cuts (Hamrawi et al., 2010) and Z-slices (Wagner and Hagras, 2010a; Christian Wagner, 2009). The ease of computation associated with the interval form of type-2 sets is the main driver for the wide usage of interval type-2 set and systems compared to the generalised form. One attempt to design general type-2 sets based on zSlices representation was proposed in (Christian Wagner, 2009) where survey data and device characteristics were used to build zSlices sets automatically. Another attempt to learn general type-2 fuzzy logic systems using alpha-plane representation has been introduced in (Mendel et al., 2009). Some other works using some neural networks concepts or classification algorithms such as: type 2 Adaptive Network Based Fuzzy Inference System (ANFIS) (John and Czarnecki, 1998), general type-2 fuzzy neural network (GT2FNN) (Jeng et al., 2009) and fuzzy C-means algorithm with a model known as “efficient triangular type-2 fuzzy logic system” (Starczewski, 2009b). To the best of the author’s knowledge, no attempt to employ a learning method on general type-2 fuzzy logic systems using the vertical-slice representation has been made. The next section will introduce the simulated annealing algorithm.
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Extending similarity measures of interval type 2 fuzzy sets to general type 2 fuzzy sets

Extending similarity measures of interval type 2 fuzzy sets to general type 2 fuzzy sets

One of the most common tools of fuzzy logic is similarity measures (SMs). A SM between FSs indicates the degree to which the FSs are similar. The concept is relevant in many fields, for example, pattern recognition [1], analogical reason- ing [2] and fuzzy rule base simplification [3]. SMs for T1 FSs have been extensively studied by many researchers, such as [4], [5] and [6] where the latter provides a good overview. However, SMs for T2 FSs have been less widespread. Al- though some methods have been developed for interval T2 FSs, e.g. [7], [8], [9], [10], [11], fewer methods exist for general T2 FSs.
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Interval type-2 intuitionistic fuzzy logic system for non-linear system prediction

Interval type-2 intuitionistic fuzzy logic system for non-linear system prediction

Abstract—This paper presents an approach to prediction based on a new interval type-2 intuitionistic fuzzy logic system (IT2IFLS) of Takagi-Sugeno-Kang (TSK) fuzzy inference. The gradient descent algorithm (GDA) is used to adapt the parame- ters of the IT2IFLS. The empirical comparison is made on the designed system using two synthetic datasets. Analysis of our results reveal that the presence of additional degrees of freedom in terms of non-membership functions and hesitation indexes in IT2IFLS tend to reduce the root mean square error (RMSE) of the system compared to a type-1 fuzzy logic approach and some interval type-2 fuzzy systems.
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Design Of High-performance Cmos Circuits For Interval Type-2 Fuzzy Logic Controller

Design Of High-performance Cmos Circuits For Interval Type-2 Fuzzy Logic Controller

ARALIK DEĞERLİ TİP-2 BULANIK MANTIK SİSTEMLERİ İÇİN YÜKSEK BAŞARIMLI CMOS DEVRE TASARIMI ÖZET Bulanık küme teorisinin ilk olarak Lotfi A. Zadeh tarafından 1965 yılında ortaya atılması ile birlikte, keskin olmayan verilerin işlenmesi için sistematik bir yöntem olan bulanık mantıksal sistemler geliştirilmiştir. Klasik mantıksal sistemlerinin temeli “yanlış” ve “doğru” elemanlarından oluşan iki değerli kümeler iken, bulanık mantık sistemleri [0, 1] kapalı aralığında sürekli değerler alabilen çok değerli sistemlerdir. [0, 1] kümesinin sınır noktaları olan 0 ve 1 değerleri, sırasıyla klasik mantıksal sistemlerindeki “yanlış” ve “doğru” durumuna karşı gelmektedir. Bu çok değerli yaklaşım gerçek dünyada karşılaşılan bilginin genelde muğlak olması, keskin olmaması nedeniyle, fiziksel sistemlerin tanımlanmasında çok yararlı olmaktadır. Bulanık mantık tabanlı denetleyici sistemler, kısaca bulanık mantık denetleyicileri olarak adlandırılmaktadır. Bulanık mantık denetleyicileri işaret işleme, örüntü tanıma, sınıflandırma ve sistem modelleme gibi farklı mühendislik sistemlerinin analizinde kullanılmaktadır. Buna karşın bulanık mantığın en önemli mühendislik uygulaması, kontrol sistemlerinde olmuştur. Farklı sistemlerin denetlenmesi için hem yazılım tabanlı, hem de donanım tabanlı farklı bulanık denetleyiciler geliştirilmiştir. Bulanık küme işlemlerinin paralel işleme ve yoğun hesaplamalara imkan veren yapısından dolayı, bu tür sistemler özel donanım kullanılarak etkin bir şekilde gerçekleştirilebilmektedir. Bu doğrultuda, bulanık denetleyicilerin yüksek başarımlı gerçeklenmeleri için, CMOS, BJT ve BiCMOS entegre devre teknolojileri ile özel tümdevre tasarımları geliştirilmiştir.
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Interval Type 2 Fuzzy Logic Tuning

for PID Controller of DC Servo Motors

Interval Type 2 Fuzzy Logic Tuning for PID Controller of DC Servo Motors

Although previous researchers suggested that controllers utilizing T1FL are capable of providing better performance, there is still room for further improvement with T2FL. This T2FL theoretically cures uncertainties more effectively but it is also more difficult to work on due to its complicated composition [8]. Recently, T2FL systems have been an attractive research area since lots of unexplored factors remain in control applications. Hence, more work is required for better understanding of T2FL performance in controlling, particularly for DC servo motors.
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“An Interval Type-2 Fuzzy Logic Approach for Induction Motors Stator Condition Monitoring”

“An Interval Type-2 Fuzzy Logic Approach for Induction Motors Stator Condition Monitoring”

Type-1 fuzzy logic controllers employ crisp and precise Type-1 fuzzy sets. For example, consider a Type-1 fuzzy set representing the linguistic label of ‘Low’ voltage. If the input voltage x is 115v, then the membership of this input to the ‘Low’ set will be the certain and crisp membership value of 0.4. However, the center and end points of this Type-1 fuzzy set can vary due to uncertainties (which could arise for example from noise) in the measurement of voltage (numerical uncertainty) and in the situations in which 115v could be called low (linguistic uncertainty). If this linguistic label was employed with a fuzzy logic controller, then the Type-1 fuzzy logic controller would need to be frequently tuned to handle such uncertainties. Alternatively, one would need to have a group of separate Type-1 sets and Type-1 fuzzy logic controllers, where each fuzzy logic
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